blem 1: Recall that binary search trees enjoy O(ig(n)) search time complexity when the tree s fairly balanced, i.e., there are no nodes where the tree grows much deeper on the left side han the right size (and vice versa). This is well depicted by a picture: BALANCED b be 0-00 UN BALANCED n this problem, we try to come up with a formula/algorithm to measure how balanced a tree is.
blem 1: Recall that binary search trees enjoy O(ig(n)) search time complexity when the tree s fairly balanced, i.e., there are no nodes where the tree grows much deeper on the left side han the right size (and vice versa). This is well depicted by a picture: BALANCED b be 0-00 UN BALANCED n this problem, we try to come up with a formula/algorithm to measure how balanced a tree is.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
![**Problem Statement and Analysis:**
**Problem 1:** Binary search trees have a logarithmic search time complexity, O(lg(n)), when the tree is balanced. A balanced tree ensures no side (left or right) of the root has nodes much deeper than the other side. This concept is illustrated by a simple diagram:
- The left diagram shows a "Balanced" binary tree.
- The right diagram depicts an "Unbalanced" binary tree.
**Objective:** Develop a formula or algorithm to measure how balanced a binary tree is.
**Definitions and Formulation:**
- **Height of a Node:** The height is defined as the maximum number of generations beneath a given node. Examples:
- A leaf node has a height of 0.
- A node with only left and/or right children and no grandchildren has a height of 1.
- A node with at least one grandchild and no great-grandchildren has a height of 2.
- Denote the height of a node \( n \) as \( h(n) \).
- **Balance of a Node:** For a node \( n \), the balance is defined as:
\[
b(n) = |h(n.left) - h(n.right)|
\]
Here, \( h(\text{NIL}) = -1 \) ensures the functionality when nodes do not exist.
- **Balance of the Tree:** The overall balance of the tree \( T \) is given by:
\[
B(T) = \max_{n \, \text{node in} \, T} b(n)
\]
**Algorithm Requirement:**
Develop an algorithm that calculates \( B(T) \) for a binary tree \( T \). The algorithm should operate in time complexity \( O(m) \), where \( m \) is the number of nodes in the tree.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2814c856-eb50-491c-9dbe-07e3e217481b%2F8da85199-84db-4813-bb42-5133a19641dc%2Ffa0c9gl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement and Analysis:**
**Problem 1:** Binary search trees have a logarithmic search time complexity, O(lg(n)), when the tree is balanced. A balanced tree ensures no side (left or right) of the root has nodes much deeper than the other side. This concept is illustrated by a simple diagram:
- The left diagram shows a "Balanced" binary tree.
- The right diagram depicts an "Unbalanced" binary tree.
**Objective:** Develop a formula or algorithm to measure how balanced a binary tree is.
**Definitions and Formulation:**
- **Height of a Node:** The height is defined as the maximum number of generations beneath a given node. Examples:
- A leaf node has a height of 0.
- A node with only left and/or right children and no grandchildren has a height of 1.
- A node with at least one grandchild and no great-grandchildren has a height of 2.
- Denote the height of a node \( n \) as \( h(n) \).
- **Balance of a Node:** For a node \( n \), the balance is defined as:
\[
b(n) = |h(n.left) - h(n.right)|
\]
Here, \( h(\text{NIL}) = -1 \) ensures the functionality when nodes do not exist.
- **Balance of the Tree:** The overall balance of the tree \( T \) is given by:
\[
B(T) = \max_{n \, \text{node in} \, T} b(n)
\]
**Algorithm Requirement:**
Develop an algorithm that calculates \( B(T) \) for a binary tree \( T \). The algorithm should operate in time complexity \( O(m) \), where \( m \) is the number of nodes in the tree.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education